EAM series and their history

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1 EM seres and ther hstory Embedded tom Method (EM and Modfed EM (MEM By Kuno TKHSHI Tokyo Insttute of Technology, Tokyo 5-855, JPN Phone/Fax

2 Quantum calculaton ( wth the smplest example Schrödnger equaton HΨ = εψ Hamltonan,, 3,,,,,,,,, N-, N Electrons Ions even f assumng only Coulomb nteracton... Wave functon ; Slater determnant Molecular orbtals must be arbtrary functon. Do you know exact soluton? -e +Z ae N N M N = + + M D e Zae H m = 4πε > = a = = a r R 0, e,,, ( ( a a b > = b Ψ( x,, x = N N! φ ( x N φ ( x φ ( x N R a, r e,, r,,,, 3,,, a,,, b,,, M-, M N φ ( x N Z a r Z b e,, a, b

3 Quantum calculaton ( How shall we approxmate? Molecular orbtals Lnear combnaton of atomc orbtals? Gaussan-type or Slater-type? How many atomc orbtals? Bass set s always effectve for any stuaton? Bass sets are optmzed to express equlbrum state. Interactons n Hamltonan Densty functonal? Whch functon? How determne?... Densty functonals are optmzed to express bulk propertes. How many electrons? Ψ( x,, x = φ ( x Egenvalue problem, the egenvalue s smlar to each other Cut off of Integraton?... ccuracy of matrx calculaton arses. N N! N φ ( x φ ( x N N φ ( x N

4 Quantum calculaton Theoretcal background Quantum calculaton to EM seres Relablty of approxmaton Inter-atomc Potental for Molecular Mechancs (Dynamcs non-quantum method Two body potental, Three body, Mut-body Usually, no theoretcal background relablty Whch wll you choose?

5 pplcablty to Large System (EM and MEM Expectable rather than quantum calc. (especally for Bulk pplcablty to Surface Problems (MEM S 7 7 DS structure Calculaton of surface energy and smulaton of reconstructon for S ( 3X3, 5X5, 7X7, and 9X9... ( Kuno TKHSHI, Chkara NR, Takahro YMGISHI, and Tadao ONZW ppled Surface Scence, vol.5,no.3-4, pp ,(999 FCC (0 Mssng Row structure Modfed embedded atom method (MEM calculatons for reconstructed (0 surfaces of face... ( Takahro YMGISHI, Kuno TKHSHI, and Tadao ONZW, Surface Scence, vol.445, no., pp.8--,(000 Quantum calculaton to EM seres ; Embedded tom Method(EM EM and Modfed EM Expectable! However, problems reman. ( ex. Pd (0 surf., clusters,,, problems nduced by the method? or the parameters?

6 (Embedded tom Method Quantum calculaton to EM EM hypothess E tot = E E Embeddng energy term + Two body term = F ( + = ( R φ ( R ( R Dfference between EM and MEM EM: sphercal symmetry ncludes s, p, d, f MEM: s, p, d, f symmetres M.W.Fnns and J.E.Snclar potental FS potental s same as EM. never takng account of s,p,d,f symmetres For non-bulk problems ex. surface, cluster, etc MEM rather than EM or FS potental pplcablty of MEM stll unknown especally for non-bulk problem + R B -... Embeddng Energy Electron gas

7 Hstory of EM seres LJ-MEM; Baskes,, Phys.Rev.Let. 83, ( Determnaton of MEM parameters for N Baskes,, Mater. Chemst. and Phys., 50, 5-58 ( tomstc calculaton of composte nterface Baskes,, J.E.ngelo, and C.L.Bsson Bsson, Modellngg Smul.. Mater. Sc. Eng.,, (994 MEM for HCP metals. and R..Johnson, Modellng Smul.. Mater. Sc. Eng.,, ( EM: a revew of theory and applcaton. M. S. Daw,, S. M. Foles and M. I. Baskes, Mater. Sc.. Rep., 9, 5-30 ( MEM for cubc mat.and mpurtes. MEM M. I. Baskes,, Phys. Rev. B, 46, ( MEM for covalent S and Ge. M. I. Baskes,, J. S. Nelson and. F. Wrght, Phys. Rev. B, 40, ( MEM for covalent S. M. I. Baskes,, Phys. Rev. Lett., 59, ( EM for fcc metals Cu, g,, u, N,, Pd, Pt and ther alloys. S. M. Foles,, M. I. Baskes and M. S. Daw, Phys. Rev. B, 33, (986. EM 984 EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. B, 9, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. Lett., 50, (983. Hstory of EM seres Theoretcal back ground and relaton wth other theores FS potental for W cluster, L.Marvlle and W.ndreon ndreon, J. Phys. Chem., 9, ( FS potental : Surface energy for bcc metal. G.J.ckland and M.W.Fnns Fnns, Phlos. Mag.., 54, ( Unversal features of the equaton of state of metals. J. H. Rose, J. R. Smth, F. Gunea and J. Ferrante, Phys. Rev. B, 9, (984. FS potental for transton metal. M.W.Fnns and J.E.Snclar, Phlos.Mag Mag.., 50, 45-55(984, 98 Effectve theory of chemcal bndng: H n 3d metals. J. K. Norskov,, Phys. Rev. B, 6, ( toms embedded n an electron gas: Immerson energes. M. J. Puska,, R. M. Nemnen and M. Mannnen, Phys. Rev. B, 4, ( Quasatoms: : atoms n nonunform electronc systems. M. J. Stott and E. Zaremba,, Phys. Rev. B,, ( Surf.energy.short-range-body potental R..Johnson and P.J.Whte, Phys. Rev.B, 3, (976

8 EM84 ( N N,, Pd, and H EM hypothess Embeddng energy term + Two body term E tot = E = F ( + Electronc densty = Supermposton of atomc densty tomc densty by Roothaan Hartree Fock E.Clement Clement,, C.Roett Roett,, tomc data and Nuclear data tables, 4, (974 (Isolated atom, wth ~4 Slater type orbals Embeddng functon E J. K. Norskov,, Phys. Rev. B, 6, (98. M. J. Puska,, R. M. Nemnen and M. Mannnen,, Phys. Rev. B, 4, (98. M. J. Stott and E. Zaremba,, Phys. Rev. B,, (980. φ ( R Par potental term φ ( = Z ( R Effectve charge dstrbuton functon determned from Lattce const., Elastc consts., Vacancy formaton energy, and heat of sublmaton R Z ( R / R ( R Brth of EM seres Hstory of EM seres LJ-MEM; Baskes,, Phys.Rev.Let. 83, ( Determnaton of MEM parameters for N Baskes,, Mater. Chemst. and Phys., 50, 5-58 ( tomstc calculaton of composte nterface Baskes,, J.E.ngelo, and C.L.Bsson Bsson, Modellngg Smul.. Mater. Sc. Eng.,, (994 MEM for HCP metals. and R..Johnson, Modellng Smul.. Mater. Sc. Eng.,, ( EM: a revew of theory and applcaton. M. S. Daw,, S. M. Foles and M. I. Baskes, Mater. Sc.. Rep., 9, 5-30 ( MEM for cubc mat.and mpurtes. M. I. Baskes,, Phys. Rev. B, 46, ( MEM for covalent S and Ge. M. I. Baskes,, J. S. Nelson and. F. Wrght, Phys. Rev. B, 40, ( MEM for covalent S. M. I. Baskes,, Phys. Rev. Lett., 59, ( EM for fcc metals Cu, g,, u, N,, Pd, Pt and ther alloys. S. M. Foles,, M. I. Baskes and M. S. Daw, Phys. Rev. B, 33, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. B, 9, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. Lett., 50, (983.

9 Supermposton of atomc densty For metals, t must be good approxmaton. Example of H + on Probablty densty dstrbuton of electron at equlbrum dstance. Bondng orbtal nt-bondng orbtal Supermposton of atomc electron dstrbuton

10 EM86 ( Cu, ( Cu, g,, u, N,, Pd, Pt, and ther alloys EM hypothess Embeddng energy term + Two body term E tot = E E = F ( + Electronc densty Supermposton of atomc densty ( s and d wth contrbuton rato n s and n d tomc densty of Roothaan Hartree Fock Rose s s unversal functon J. H. Rose, J. R. Smth, F. Gunea and J. Ferrante, Phys. Rev. B, 9, (984. Embeddng functon Par potental term Effectve charge dstrbuton functon Evoluton of EM seres ( φ ( R φ ( R = Z ( R Z ( R / R = ( R Hstory of EM seres LJ-MEM; Baskes,, Phys.Rev.Let. 83, ( Determnaton of MEM parameters for N Baskes,, Mater. Chemst. and Phys., 50, 5-58 ( tomstc calculaton of composte nterface Baskes,, J.E.ngelo, and C.L.Bsson Bsson, Modellngg Smul.. Mater. Sc. Eng.,, (994 MEM for HCP metals. and R..Johnson, Modellng Smul.. Mater. Sc. Eng.,, ( EM: a revew of theory and applcaton. M. S. Daw,, S. M. Foles and M. I. Baskes, Mater. Sc.. Rep., 9, 5-30 ( MEM for cubc mat.and mpurtes. M. I. Baskes,, Phys. Rev. B, 46, ( MEM for covalent S and Ge. M. I. Baskes,, J. S. Nelson and. F. Wrght, Phys. Rev. B, 40, ( MEM for covalent S. M. I. Baskes,, Phys. Rev. Lett., 59, ( EM for fcc metals Cu, g,, u, N,, Pd, Pt and ther alloys. S. M. Foles,, M. I. Baskes and M. S. Daw, Phys. Rev. B, 33, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. B, 9, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. Lett., 50, (983.

11 MEM9 ( Cu, ( Cu, g,, u, N,, Pd, Pt, l, Pb, Rh, Ir, L, Na,, K, V, Nb, Ta, Cr, Mo,, W, Fe, C, S, Ge,, H, N, O and ther compounds EM hypothess Embeddng energy term + Two body term E tot = E = F ( + Embeddng functon Fxed F ( = Electronc densty Supermposton of atomc densty s, p, d, f symmetry Parameterzed atomc densty Par potental term a( l Reference structure Rose s s unversal functon J. H. Rose, J. R. Smth, F. Gunea and J. Ferrante, Phys. Rev. B, 9, (984. Screenng functon to neglect farther atoms? E Evoluton of EM seres ( φ ( R E0 ln ( l ( l ( = t ( l= 0 ( l R = exp β 0 R Hstory of EM seres LJ-MEM; Baskes,, Phys.Rev.Let. 83, ( Determnaton of MEM parameters for N Baskes,, Mater. Chemst. and Phys., 50, 5-58 ( tomstc calculaton of composte nterface Baskes,, J.E.ngelo, and C.L.Bsson Bsson, Modellngg Smul.. Mater. Sc. Eng.,, (994 MEM for HCP metals. and R..Johnson, Modellng Smul.. Mater. Sc. Eng.,, ( EM: a revew of theory and applcaton. M. S. Daw,, S. M. Foles and M. I. Baskes, Mater. Sc.. Rep., 9, 5-30 ( MEM for cubc mat.and mpurtes. M. I. Baskes,, Phys. Rev. B, 46, ( MEM for covalent S and Ge. M. I. Baskes,, J. S. Nelson and. F. Wrght, Phys. Rev. B, 40, ( MEM for covalent S. M. I. Baskes,, Phys. Rev. Lett., 59, ( EM for fcc metals Cu, g,, u, N,, Pd, Pt and ther alloys. S. M. Foles,, M. I. Baskes and M. S. Daw, Phys. Rev. B, 33, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. B, 9, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. Lett., 50, (983.

12 Embeddng functon n MEM9 F( = ln( Fxed formalsm = F ( E0 ln 0 0 at equlbrum 0 = Functon F( Equlbrum by Baskes dfferent from Equlbrum of EM Physcal meanng decreased Parameterzed Parameter for stackng fault energy (FCC, HCP Equlbrum by Johnson rgument of functon Embeddng term can be covered by par-potental term

13 Electronc densty n MEM9 Supermposton of atomc densty s, p, d, f symmetry s ( l=0 (0 = a(0 ( R E tot = E E = F ( + φ ( R 3 ( l ( l ( = t ( l= 0 ( l (0 ( / 3 (0 ( l = + t l= p ( l= d ( l= f ( l=3 ( ( = ( ( α x α a( ( R ( α β a = x x ( R 3 α, β (3 ( = α, β, γ x α β γ x x a(3 ( R α = a( α x R / R ( R + R B -... Parameterzed atomc densty a( l ( l R = exp β 0 R parameters ; 0 R (l t (l β

14 bout s, p, d, f symmetry n MEM9 l θ l Usual expresson ( ust n case s (0 ( =, l( a(0 ( R a(0 l ( R l (0 = a(0 ( R R l R p ( ( a( ( = ( R ( R cos( θ, l( a l l l ( ( = α x α a( ( R d ( ( a( ( = ( R ( R cos ( θ, l( a l l l 3 ( ( ( α β a = x x ( R 3 α, β a( ( R f (3 (3 a(3 3 ( = ( R ( R cos ( θ, l( a l l l (3 ( = α, β, γ x α β γ x x a(3 ( R Supermposton of no wave functon but electronc densty nt-bondng state would be never expressed. a( l ( l R = exp β 0 R

15 pplcablty of MEM9 Stll unknown Expectable rather than quantum calculaton. Not only calculaton tme but also relablty n approcmaton ccuracy must be examned. ccuracy of both Method and Parameters Bulk problems Surface problems to cluster problems S 7 7 DS structure Calculaton of surface energy and smulaton of reconstructon for S ( 3X3, 5X5, 7X7, and 9X9... ( Kuno TKHSHI, Chkara NR, Takahro YMGISHI, and Tadao ONZW ppled Surface Scence, vol.5,no.3-4, pp ,(999 FCC (0 Mssng Row structure Modfed embedded atom method (MEM calculatons for reconstructed (0 surfaces of face... ( Takahro YMGISHI, Kuno TKHSHI, and Tadao ONZW, Surface Scence, vol.445, no., pp.8--,(000

16 MEM9 applcaton to S 7x7 DS structure ( DS structure ; K.Takayanag, Y.Tanshro, M.Takahash, and S.Takahash, J. Vac. Sc. Technol., 3, 50 (985 Why 7x7 s observed. Sometmes 5x5 s observed rather than 9x9. Dmmer datom Rest atom Corner home

17 MEM9 applcaton to S 7x7 DS structure ( Kuno TKHSHI, Chkara NR, Takahro YMGISHI, and Tadao ONZW, ppled Surface Scence, vol.5,no.3-4, pp ,(999 7x7 s stable. 5x5 s more stable than 9x9. Frst prncple calculaton : I.Stch, M.C.Payne, R.D.Kng-Smth, J-S.Ln, and L.J.Clarke, Phys. Rev. Lett., 68, 35 (99. Tght bndng estmaton : Y.F.Zhao, H.Q.Yang, J.N.Gao, Z.Q.Xue, and S.J.Pang, Phys. Rev. B, 58, 384 (998. Surface energy (mj/m present study Reference Structure before after Stch Zhao reconstructon reconstructon

18 MEM9 applcaton to FCC(0 structure ( Takahro YMGISHI, Kuno TKHSHI, and Tadao ONZW, Surface Scence, vol.445, no., pp.8--,(000 FCC mssng row structure Ideal Mssng row Mssng column [0] [0] [00] (a (b (c Sude vew Top vew

19 MEM9 applcaton to FCC(0 structure ( Results by MEM9 Materal Surface Energy (mj/m 3 Ideal M.R. M.C. Stable structure ( exp. Good agreement must be careful Problem remans method or parameter u M.R. Pt M.R. Pb Ideal Rh Ideal Pd Ideal N ? [0] [0] [00] (a (b (c Sude vew Top vew Takahro YMGISHI, Kuno TKHSHI, and Tadao ONZW, Surface Scence, vol.445, no., pp.8--,(000

20 Results by MEM9 MEM9 applcaton to FCC(0 structure (3 Good agreement must be careful [00] 4 Problem remans method or parameter Takahro YMGISHI, Kuno TKHSHI, and Tadao ONZW, Surface Scence, vol.445, no., pp.8--,(000 [0] 3 Element 3 4 Drecton [00] [0] [00] [0] [00] [0] [00] [0] MEM u LEED[9] Pt MEM LEED[0]

21 pplcablty of MEM9 Stll unknown Probably the best method for very large bulk Maybe useful for large surface problem Be careful for cluster problem Expectable rather than quantum calculaton. For large system Not only calculaton tme but also relablty n calculaton ccuracy must be examned. ccuracy of Method Parameters

22 LJ-EM99 EM hypothess E tot = E = F( + No s,p,d,f symmetry no dfference between FCC and HCP Rose s s functon -> Lennard Jones (LJ Par potental term s determned so that the results are same as LJ. Unversal feature of materal? E Evoluton of EM seres (3 φ ( R F( = E0 ln 0 0 = ( R R ( R = exp β R 0 β Hstory of EM seres LJ-MEM; Baskes,, Phys.Rev.Let. 83, ( Determnaton of MEM parameters for N Baskes,, Mater. Chemst. and Phys., 50, 5-58 ( tomstc calculaton of composte nterface Baskes,, J.E.ngelo, and C.L.Bsson Bsson, Modellngg Smul.. Mater. Sc. Eng.,, (994 MEM for HCP metals. and R..Johnson, Modellng Smul.. Mater. Sc. Eng.,, ( EM: a revew of theory and applcaton. M. S. Daw,, S. M. Foles and M. I. Baskes, Mater. Sc.. Rep., 9, 5-30 ( MEM for cubc mat.and mpurtes. M. I. Baskes,, Phys. Rev. B, 46, ( MEM for covalent S and Ge. M. I. Baskes,, J. S. Nelson and. F. Wrght, Phys. Rev. B, 40, ( MEM for covalent S. M. I. Baskes,, Phys. Rev. Lett., 59, ( EM for fcc metals Cu, g,, u, N,, Pd, Pt and ther alloys. S. M. Foles,, M. I. Baskes and M. S. Daw, Phys. Rev. B, 33, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. B, 9, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. Lett., 50, (983.

23 LJ-EM99 Unversal feature of materal? β No s,p,d,f symmetry 8 4 FCC BCC no dfference between FCC and HCP SC Possblty of extenson Dmer Graphte DI β Hstory of EM seres LJ-MEM; Baskes,, Phys.Rev.Let. 83, ( Determnaton of MEM parameters for N Baskes,, Mater. Chemst. and Phys., 50, 5-58 ( tomstc calculaton of composte nterface Baskes,, J.E.ngelo, and C.L.Bsson Bsson, Modellngg Smul.. Mater. Sc. Eng.,, (994 MEM for HCP metals. and R..Johnson, Modellng Smul.. Mater. Sc. Eng.,, ( EM: a revew of theory and applcaton. M. S. Daw,, S. M. Foles and M. I. Baskes, Mater. Sc.. Rep., 9, 5-30 ( MEM for cubc mat.and mpurtes. M. I. Baskes,, Phys. Rev. B, 46, ( MEM for covalent S and Ge. M. I. Baskes,, J. S. Nelson and. F. Wrght, Phys. Rev. B, 40, ( MEM for covalent S. M. I. Baskes,, Phys. Rev. Lett., 59, ( EM for fcc metals Cu, g,, u, N,, Pd, Pt and ther alloys. S. M. Foles,, M. I. Baskes and M. S. Daw, Phys. Rev. B, 33, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. B, 9, ( EM for H n Metals. M. S. Daw and M. I. Baskes, Phys. Rev. Lett., 50, (983.

Set the initial conditions r i. Update neighborlist. r i. Get new forces F i

Set the initial conditions r i. Update neighborlist. r i. Get new forces F i Set the ntal condtons r t 0, v t 0 Update neghborlst Potental models for metals Get new forces F r Solve the equatons of moton numercally over tme step t : r t n r t n + v t n v t n + Perform T, P scalng